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 Friday Seminar on Knot Theory（2010） (2009) (2011)
Organizer in 2010 : IWAKIRI, Masahide
Speaker ：Sumiko Horiuchi (Tokyo Woman's Christian University)
Title ：A two dimensional lattice of knots by $C_{2n}$-moves
（Abstract） （PDF）
Date ：January 28 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Sang Youl Lee (Pusan National University)
Title ：On Tripp's conjecture about the canonical genus
for Whitehead doubles of alternating knots
（Abstract） （PDF）
Date ：January 28 (Fri.) 14：30～15：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kumi Kobata (OCAMI, Osaka City University)
Title ：A generalization of an enumeration on self-complementary graphs
for edge colored graphs
（Abstract） （PDF）
Date ：January 28 (Fri.) 13：00～14：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Taizo Kanenobu (Osaka City University)
Title ：Ribbon torus knots presented by virtual knots with up to 4 crossings
（Abstract） （PDF）
Date ：January 28 (Fri.) 10:30～11：30
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Hiromasa Moriuchi (OCAMI, Osaka City University)
Title ：Covering link polynomials for generalized Kinoshita's theta-curve
（Abstract） （PDF）
Date ：January 21 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Takefumi Nosaka (RIMS, Kyoto University)
Title ：Mochizuki $3$-cocycle invariant of links
in $S^3$ is one of Dijkgraaf-Witten invariant
（Abstract） （PDF）
Date ：December 17 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Masahide Iwakiri (OCAMI, Osaka City University)
Title ：Surface-links represented by 4-charts and quandle cocycle invariants II
（Abstract） （PDF）
Date ：December 3 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Sakie Suzuki (RIMS, Kyoto University)
Title ：On the universal invariant of bottom tangles
（Abstract） （PDF）
Date ：November 26 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Inasa Nakamura (RIMS, Kyoto University)
Title ：Quandle cocycle invariant of a certain T^2-link
（Abstract） （PDF）
Date ：November 19 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Yoshiro Yaguchi (Hiroshima University)
Title ：Infinite type invariant of surface braids with 4 branch points
（Abstract） （PDF）
Date ：November 12 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Shin Satoh (Kobe University)
Title ：Fox colorings and cocycle invariants of roll-spun knots
(joint work with Masahide Iwakiri (OCAMI))
（Abstract） （PDF）
Date ：November 5 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kengo Kishimoto (Osaka City University, OCAMI)
Title ：A relation between sharp move and Delta move
（Abstract） （PDF）
Date ：October 29 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Kazuto Takao (Osaka University)
Title ：Normalization of the Rubinstein-Scharlemann graphic of Morse functions
（Abstract） （PDF）
Date ：October 8 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Taizo Kanenobu (Osaka City University)
Title ：Finite type invariants for a spatial handcuff graph
（Abstract） （PDF）
Date ：June 11 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ayako Ido (Nara Women's University)
Title ：Rubinstein-Scharlemann graphic of 3-manifolds and Hempel distance
（Abstract） （PDF）
Date ：May 28 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Koki Masumoto (Osaka University)
Title ：On the hyperbolic volume of PSL(2,C)-representations
of fundamental groups
（Abstract） （PDF）
Date ：May 21 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Shin’ya Okazaki (Osaka City University)
Title ：On a homeomorphism obtained by bridge position of a knot
（Abstract） （PDF）
Date ：May 14 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Noboru Ito(Waseda University)
Title ：Chain homotopy maps for Khovanov homology
（Abstract） （PDF）
Date ：May 7 (Fri.) 16：10～17：10
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Teruhisa Kadokami (East China Normal University)
Title ：Properties of Gauss phrase and category of regions
(with Yusuke Kiriu (Studio Phones))
（Abstract） （PDF）
Date ：May 7 (Fri.) 15：00～16：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Ayaka Shimizu　(Osaka City University)
Title ：On the distribution of the ordered linking warping degree
（Abstract） （PDF）
Date ：April 23 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Naoyuki Monden　(Osaka University)
Title ：Generating sets of the mapping class group
（Abstract） （PDF）
Date ：April 16 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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Speaker ：Masahide Iwakiri (OCAMI, Osaka City University)
Title ：Infinite sequences of mutually non-conjugate surface braids representing
（Abstract） （PDF）
Date ：April 9 (Fri.) 16：00～17：00
Place ：Dept. of Mathematics, Sci. Bldg., 3153
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 Abstracts

 Speaker： Sumiko Horiuchi (Tokyo Woman's Christian University) Title： A two dimensional lattice of knots by $C_{2n}$-moves

We consider a local move on a knot diagram, where we denote the local move by $M$.
If two knots $K_1$ and $K_2$ are transformed into each other by a finite sequence of $M$-moves, the $M$-distance between $K_1$ and $K_2$ is the minimum number of times of $M$-moves needed to transform $K_1$ into $K_2$. A $M$-distance satisfies the axioms of distance.
A two dimensional lattice of knots by $M$-moves is the two dimensional lattice graph which satisfies the following : The vertex set consists of oriented knots and for any two vertices $K_1$ and $K_2$, the distance on the graph from $K_1$ to $K_2$ coincides with the $M$-distance between $K_1$ and $K_2$, where the distance on the graph means the number of edges of the shortest path which connects the two knots.
Local moves called $C_n$-moves are closely related to Vassiliev invariants.
In this talk, we show that for any given knot $K$, there is a two dimensional lattice of knots by $C_{2n}$-moves $(n>1)$ with the vertex $K$.

 Speaker： Sang Youl Lee (Pusan National University) Title： On Tripp's conjecture about the canonical genus for Whitehead doubles of alternating knots

In 2002, J. Tripp conjectured that the minimal crossing number of a knot coincides with the canonical genus of its Whitehead double. In this talk, I'd like to introduce a family of alternating knots for which Tripp's conjecture holds.

 Speaker： Kumi Kobata (OCAMI, Osaka City University) Title： A generalization of an enumeration on self-complementary graphs for edge colored graphs

We plan to give a generalization of Tazawa-Ueno's theorem.
Tazawa-Ueno's theorem is a formula for enumeration on self-complementary bipartite graphs with given number of vertices. We generalize the enumeration to edge colored bipartite graphs. We also consider the case of edge colored digraphs as well as ordinary graphs. This talk is a joint work with Yasuo Ohno.

 Speaker： Taizo Kanenobu (Osaka City University) Title： Ribbon torus knots presented by virtual knots with up to 4 crossings

A ribbon torus knot embedded in the 4-space is presented by a welded virtual knot through the Tube operation due to Shin Satoh.
We make an attempt of classification of ribbon torus knots presented by virtual knots with up to 4 crossings, where we use the list of virtual knots enumerated by Jeremy Green. We make use of the classification of the groups of the virtual knots with up to 4 crossings due to Atsushi Ichimori.

 Speaker： Hiromasa Moriuchi (OCAMI, Osaka City University) Title： Covering link polynomials for generalized Kinoshita's theta-curve

Kinoshita's theta-curve $\theta(1,1,1)$ is almost unknotted theta-curve, that is, its constituent knots are all trivial. We consider generalized Kinoshita's theta-curve $\theta(i,j,k)$ by adding full-twists. In this talk, we introduce covering link polynomials to classify $\theta(i,j,k)$. We also mention influence on order-3 vertex connected sum of $\theta(i,j,k)$ and $\theta(i',j',k')$.

 Speaker： Takefumi Nosaka (RIMS, Kyoto University) Title： Mochizuki $3$-cocycle invariant of links in $S^3$ is one of Dijkgraaf-Witten invariant

Let p be an odd prime, and $\phi$ the Mochizuki $3$-cocycle of the dihedral quandle of order $p$. Using $\phi$, Carter-Kamada-Saito combinatorially defined a shadow cocycle invariant of links in $S^3$. Let $M_L$ be the double covering branched along a link L. Our main result is that the cocycle invariant of L is equal to the Dijkgraaf-Witten invariant of $M_L$ with respect to $Z/pZ$ up to scalar multiples. We further compute Dijkgraaf-Witten invariants of some $3$-manifolds. In this talk, we present a simple proof of the equality. This work is joint with Eri Hatakenaka.

 Speaker： Masahide Iwakiri (OCAMI, Osaka City University) Title： Surface-links represented by 4-charts and quandle cocycle invariants II

In this talk, we study surface-links represented by 4-charts and their dihedral quandle cocycle invariants. As a consequence, we characterize 4-charts representing some surface-links including a 2-twist spun trefoil. We also prove that the braid index of a connected sum of a 2-twist spun trefoil and a spun trefoil is five, which is the answer to a special case of Tanaka's Problem.

 Speaker： Sakie Suzuki (RIMS, Kyoto University) Title： On the universal invariant of bottom tangles

bottom tangle is a tangle in a cube whose boundary points are arranged on the bottom line, and every link can be obtained from a bottom tangle by closing. The universal sl_2 invariant of bottom tangles has the universality property with respect to the colored Jones polynomial of links. In this talk, we study the universal sl_2 invariant of certain types of bottom tangles.

 Speaker： Inasa Nakamura (RIMS, Kyoto University) Title： Quandle cocycle invariant of a certain T^2-link

We consider a surface link which is presented by a simple branched covering over the standard torus, which we call a torus-covering link. A torus-covering $T^2$-link is determined from two commutative classical $m$-braids, which we call basis $m$-braids, and we denote by $\mathcal{S}_m(a,b)$ the torus-covering $T^2$-link with basis $m$-braids $a$ and $b$. In this talk we present the quandle cocycle invariant of $\mathcal{S}_m(b, \Delta^{2n})$, by using the quandle cocycle invariants of the closure of $b$, where $\Delta$ is a half twist of a bundle of $m$ parallel strands.

 Speaker： Yoshiro Yaguchi (Hiroshima University) Title： Infinite type invariant of surface braids with 4 branch points

Hurwitz action of the n braid group $B_n$ on the n-fold direct product of a group $G$ is studied. Hurwitz action can be used in study of surface braids. In this talk, we will give an infinite type invariant of surface braids with 4 branch points by determining the orbit decomposition of Hurwitz action on $G^4$ when $G$ is the semi-direct product $S_m\ltimes Z^m$, where $S_m$ is the symmetric group of degree m and $Z^m$ is the $m$-fold direct product of the cyclic group $Z$.

 Speaker： Shin Satoh (Kobe University) Title： Fox colorings and cocycle invariants of roll-spun knots (joint work with Masahide Iwakiri (OCAMI))

A roll-spun knot is a knotted 2-sphere in 4-space obtained by spinning a classical knot with rollings along the longitude. We show how to calculate the cocycle invariant of a roll-spun knot generally, and prove that the invariant of any roll-spun knot is always trivial in the case of the dihedral quandle.

 Speaker： Kengo Kishimoto (Osaka City University, OCAMI) Title： A relation between sharp move and Delta move

In this talk, we study certain local moves for knots. First we observe a relation between the sharp move and the Delta move. We show that the sharp-unknotting number of a knot is less than three times the Delta-unknotting number of the knot. Second we consider a property of the sharp-Gordian graph. The sharp-Gordian graph $\mathcal{G}_{\#}$ is a bipartite graph because a single sharp move changes the arf invariant. We show that, for any knot and any natural numbers $m,n$, there exists a complete bipartite graph $K_{m,n} \subset \mathcal{G}_{\#}$ such that $K_{m,n}$ contains the knot.

 Speaker： Kazuto Takao (Osaka University) Title： Normalization of the Rubinstein-Scharlemann graphic of Morse functions

The Rubinstein-Scharlemann graphic was introduced for studying Heegaard splittings and it has made a remarkable contribution to the
recent development of this branch. However, the graphic is constructed through a pair of smooth functions on the 3-manifold and so has much ambiguity. To extract the maximum imformation from the graphic, we have to understand how the graphic can be changed by deforming these functions. In this talk, we collect local moves on the graphic realized by deforming the functions and take an approach to the normalization of the graphic.

 Speaker： Taizo Kanenobu (Osaka City University) Title： Finite type invariants for a spatial handcuff graph

We first explain a finite type invariant, or Vassliev invariant, for a knot. Then we consider a finite type invariant for an embedded handcuff graph in a 3-sphere: We express a basis for the vector space of finite type invariants of order less than or equal to three for a spatial handcuff graph in terms of the linking number, the Conway polynomial, and the Jones polynomial of the sublinks of the handcuff graph.

 Speaker： Ayako Ido (Nara Women's University) Title： Rubinstein-Scharlemann graphic of 3-manifolds and Hempel distance

Graphic is introduced by Rubinstein-Scharlemann for studying strongly irreducible Heegaard splittings, and Kobayashi-Saeki showed that graphics can be regarded as the images of the discriminant sets of stable maps from the 3-manifolds into the plane.
In this talk, we give an introductory talk for the graphic, and also give a talk on an application of it, that is, we give a method to estimate Hempel distances.

 Speaker： Koki Masumoto (Osaka University) Title： On the hyperbolic volume of PSL(2,C)-representations of fundamental groups

Let M be a cusped 3-manifold. For a PSL(2,C)-representation of the fundamental group of M , we can define the hyperbolic volume.
In this talk we will show that the hyperbolic volume of a representation is invariant by a mutation.

 Speaker： Shin’ya Okazaki (Osaka City University) Title： On a homeomorphism obtained by bridge position of a knot

For a knot in bridge position of the three sphere, we have a Heegaard splitting of the three sphere such that the knot is included standardly in one of the Heegaard handlebodies. Then we obtain a Heegaard splitting of the zero surgery manifold along the knot from the Heegaard splitting of the three sphere.
In this talk, we consider how a Heegaard surface homeomorphism of this Heegaard splitting of the zero surgery manifold is obtained from the Heegaard splitting of the three sphere by the zero surgery of the knot. We show that a Heegaard surface homeomorphism is represented by a certain product of the generators of the mapping class group of the Heegaard surface.

 Speaker： Noboru Ito(Waseda University) Title： Chain homotopy maps for Khovanov homology

Khovanov homology is a categorification of the Jones polynomial of links.
As written in Viro's paper (O.Viro, Fund. Math. 184 (2004), 317--342), "the most fundamental property of the Khovanov homology groups is their invariance under Reidemeister moves". Khovanov homology groups are knot invariants because these groups are invariant under three types of Reidemeister moves. By giving explicit chain homotopy maps using Viro's definition of the homology, he proved the invariance under the first Reidemeister moves. This talk gives chain homotopy maps ensuring the invariance under the other Reidemeister moves. We also discuss a good property of the explicit chain homotopy maps.

 Speaker： Teruhisa Kadokami (East China Normal University) Title： Properties of Gauss phrase and category of regions (with Yusuke Kiriu (Studio Phones))

A Gauss phrase is a totally ordered $2n$ letters in which the same letter appears twice. It is described by a map $w$ from the ordinary ordered set $\{1, 2, ..., 2n\}$ to an unordered set $\{1, 2, ..., n\}$. We note that the order of the former set is important, and the latter set is just a labelling set (i.e.\ we can take the latter set like $\{ dog, cat, ..., bird\}$). By dividing the former set into $m$ parts, we obtain an $m$-component Gauss phrase. The case $m=1$ is said a Gauss word. An element of the latter set is said a crossing. If we introduce the sign $+/-$ into every crossing, then we obtain a signed Gauss phrase, where the sign corresponds to the local intersection number of two arcs near the crossing from the parameter. By closing the ends, we obtain an $m$-component flat virtual link diagram. So we can say that a Gauss phrase is a pre-flat virtual link diagram".
By introducing over/under information to every crossing of a signed Gauss phrase, we obtain a virtual link diagram. You can introduce some other structures (for example, Reidemeister equivalence) according to your purpose.

Firstly, we study properties of a Gauss phrase. As an example, we study checkability of a diagram. The property concerns deeply with an alternating virtual link. In particular, we discuss about uniqueness of minimal genus assignment for special cases, and non-classicality of one-virtualized diagram.

Secondly, from the techniques above, we discuss about operations among the complement regions of a diagram, and we define a category of regions. The concept is a dual of nanoword theory. This kind point of view has been already applied in proving Tait's flyping conjecture and in 2-knot theory.
We will apply for wider areas in the future study.

 Speaker： Ayaka Shimizu　(Osaka City University) Title： On the distribution of the ordered linking warping degrees

The ordered linking warping degree ld(D) of an ordered link diagram D is the number of non-self warping crossing points of D.
The arithmetic mean of ld(D) for all orders depends only on the non-self crossing number of D.
The standard deviation of ld(D) for all orders is zero if and only if D is in equilibrium.

 Speaker： Naoyuki Monden　(Osaka University) Title： Generating sets of the mapping class group

It is a classical problem in group theory to find small generating sets and torsion generating sets. We will consider this problem for the mapping class group of a closed orientable surface.
In this talk, we will introduce some generating sets of the mapping class group. In addition, we will show that the mapping class group is generated by 3 elements of order 3.

 Speaker： Masahide Iwakiri (OCAMI, Osaka City University) Title： Infinite sequences of mutually non-conjugate surface braids representing same surface-links

In this talk, we give an infinite sequence of mutually non-conjugate surface braids with same degree representing the trivial surface-link with at least two components and a pair of non-conjugate surface braids with same degree representing a spun (2,t)-torus knot for t≧3.
To give these examples, we introduce new invariants of conjugacy classes of surface braids via colorings by Alexander quandles or core quandles of groups.